Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind

September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Tuesday September 20, 2005
Learning Log Number 12

5:15 am Lethbridge, Alberta

I am still having difficulty finding time for Learning. I continually over-estimate how much time I plan to devote to this activity. Thus I end up with more books than I can possibly read slowly before I go on another buying spree. I bought one more math book (this time on the origins of group theory) while in Edmonton:

  • The Equation That Couldn't Be Solved (2005) by Mario Livio


Both Spivak and Courant & Robbins begin with a discussion of number. I am still struggling with finding an appropriate level of detail for studying number, before I move on to functions and limits.

I have skim read the first chapter of Courant & Robbins.


I have just looked up transcendental number in the MathWords web site. They are real numbers that are not algebraic (i.e. they are not the solution to a polynomial equation with integer coefficients). There is a superb diagram on this web page that shows visually that each of the following is a proper subset of those below it:

  • natural numbers
  • whole numbers
  • integers
  • rationals
  • algebraic
  • reals
  • imaginary
  • complex

Courant & Robbins begin with a chapter on the natural numbers and then add a substantial supplement on number theory. The second chapter deals with all of the other types of numbers. There are then three chapters on geometry and topology (which I think I can omit, at least for the moment), and then there is a chapter on functions and limits.

Regardless of whether I refer to Spivak or Courant & Robbins I have a substantial amount of work before I get to calculus. Fair enough!

Although the "counting numbers" (also called the Natural numbers) are familiar to everyone, it is only in the last century or so that we have developed a rigorous theoretical foundation for them. We can now state this foundation in terms of a few basic axioms and can then derive the remaining familiar properties using basic logic. The difficulty is not so much in the proofs, as it is in keeping track of exactly what has been proven so far, so one can use those theorems to prove additional theorems. For example, the familiar fact that the product of two negative numbers is a positive number is a logical consequence of these basic axioms.

We can take as a given the familiar properties of commutativity of addition and multiplication as well as the distributive property of multiplication over addition. The inverse operations of subtraction and division are easily defined as is the concept of power (exponent), although this latter concept needs elaboration in the case where the exponent is not an integer.

One new, but familiar concept is that of absolute value. The symbol for this concept is a pair of vertical lines on each side of the number. It is defined as follows:

"In general, the most straightforward approach to any problem involving absolute values requires treating several cases separately, since absolute values are defined by cases to begin with." [p. 11]

I am going to create a table that captures the essential concepts of chapter 2 of Spivak:

Natural numbers
The positive integers
mathematical induction
Suppose P(x) means that property P holds for the natural number x. Then P(x) is true for all x if
P(1) is true, and
whenever P(k) is true, P(k+1) is true.
well-ordering principle

If A is a nonnull set of natural numbers, then A has a least member.

The principle of mathematical induction may be proven from the well-ordering principle or vice-versa.

recursive definition
A definition that is true for n = 1 and the value for n+1 is defined in terms of n.
positive and negative integers as well as zero. Z is short for the German word "Zahl", meaning number.
Rational numbers
the ratio m/n of two integers, n not equal to 0. Q is short for quotient.
Real numbers
numbers that can be represented by infinite non-repeating decimals.
prime number
a natural number having only itself and 1 as factors
Theorem: There are an infinite number of prime numbers.
Euclid's proof by contradiction.
There is an entire branch of mathematics devoted to theorems about prime numbers.
Fibonnaci sequence

There is an entire branch of mathematics devoted to the properties of this sequence.

Done. Now to have a close look at Courant & Robbins.

Here are a few quotes from "Calculus" by Michael Spivak
"In the next chapter the deficiencies of properties P1-P12 will become quite clear, but the proper means for correcting these deficiencies is not so easily discovered. The crucial additional basic property of numbers which we are seeking is profound and subtle, quite unlike P1-P12. The discovery of this crucial property will require all of the the work of Part II of this book.

At this point I have no idea what this additional basic property might be.

Part II "Foundations" covers pages 39 - 144, so this property is obviously not trivial.


7:25 am

Total elapsed time: 2 hr. 10 min

It remains to be seen if I can have another session today....

Total elapsed time for the day: 2 hr. 10 min